Hermite's Constant for Quadratic Number Fields

Abstract

We develop a method to compute the Hermite-Humbert constants γ_K,n of a real quadratic number field K, the analogue of the classical Hermite constant γ_n when Q is replaced by a quadratic extension. In the case n = 2, the problem is equivalent to the determination of lowest points of fundamental domains in H² for the Hilbert modular group over K, that had been studied experimentally by H. Cohn. We establish the results he conjectured for the fields Q(√2), Q(√3) and Q(√5). The method relies on the characterization of extreme forms in terms of perfection and eutaxy given by the second author in an earl ier paper.